(1) Dimension, Unit, and Vector Analysis

PAGE EDITORS: Julia Telischi, Alix Hamilton, Jade Azari, and Phil Straus

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Natland Note: (9/12/13) The video still does not work! Somebody needs to see me about this one.

Your wiki page is due by 9/06/13

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NOTES:

DIMENSIONAL ANALYSIS
There are three basic types of measurements (for now).

Physical Quality	Dimensions	Base Unit
Length	[L]	meter (m)
Mass	[M]	kilogram (kg)
Time	[T]	second (s)

Other things can be measured by manipulating these dimensions.
For example:

Derived Units	Dimensions	Base Unit
Volume	[L]^3	m^3
Velocity	[L]/[T]	m/s
Acceleration	[L]/[T]^2	m/s^2

In order for an equation to be dimensionally correct, it must have the exact same dimensions on both sides.
Examples:
x = x + vt + .5at^2
L = L + (L/T)(T) + (L/T^2)(T^2)
L = L + L + L
Dimensionally Correct
(note: the coefficient .5 does not have a dimension)

v = v + 3at^2
L/T = L/T + (L/T^2)(T^2)
L/T = L/T + L
NO!

VECTORS

• Vector quantities vs. Scalar quantities
  • scalar- a physical quantity that requires only magnitude and units to be fully defined
    • ex.- mass, time, speed, length, temperature
  • vector- a physical quantity that requires magnitude AND direction to be fully defined
    • ex.- displacement, velocity, acceleration, force
• Identifying vectors
  • aren't vectors MARvelous? M=magnitude A=angle R=reference must be stated when identifying a vector
  • ex. A= 20m at 30 degrees West of North

COMPONENTS

• A component literally means an ingredient or part, so naturally they are the parts that make up a vector
• a projection of a vector onto an axis
• If the angle with the x-axis is greater than 45 degrees, the x component will be longer than the y component
• Can be thought of as shining a flashlight on the vector and seeing its shadow on the axis
• The x and y components are perpendicular to each other and form a right triangle with the resultant vector

ADDING & SUBTRACTING VECTORS

• note: adding vectors is commutative but subtracting them is not

1. Parallelogram Method

• put the two vectors being added tail-to-tail and then draw them again head-to-head to make a parallelogram
• Draw the resultant vector from the two tails to the two heads
• ex.2. Tail-to-head

Item Picture

2. Tail-to-head

• put the vectors being added tail to head and draw the resultant vector from the tail of the first vector to the head of the last vector
• To find the magnitude of the resultant vector,
  • break the vectors being added into x and y components using trig functions
  • add all of the x components, then all the y components
    • don't forget to subtract if the component vector goes in the negative direction
  • use pythagorean theorem to find the resultant magnitude
• Use inverse trig functions to find the angle of the resultant vector

• 
  
• note: "Vectors in the same direction can be added or subtracted by adding or subtracting their magnitudes. If you add two vectors in opposite directions, their magnitudes are subtracted, not added."
• 
  

SUBTRACTING VECTORS:

• Can be seen as adding the "negative" vector
• The negative of a vector has the same magnitude, but is antiparallel - that is, facing exactly 180 degrees in the opposite direction
• Either the parallelogram or the tail-to-head method can be used to calculate the resultant vector

**Quick Review of Trig Functions:

• useful for finding the x and y components of a vector as well as its angle

Example showing how to correctly apply Trig Functions:




ORDERS OF MAGNITUDE

"Powers of Ten"
http://www.youtube.com/watch?v=0fKBhvDjuy0
Scale of the Universe

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SAMPLE PROBLEMS:

1. What is the magnitude and direction of the vector on the left?
Answer: 2.3 units at 55 degrees North of East or 2.3 units at 35 degrees East of North

More practice on identifying vectors: http://www.mathwarehouse.com/vectors/


More practice on adding vectors: http://www.physicsclassroom.com/morehelp/vectaddn/practice.cfm
http://www.physicsclassroom.com/Class/vectors/u3l1b.cfm

More practice on adding vectors (word problems):
http://physics.info/vector-addition/problems.shtml


Practice on subtracting vectors (with video demonstrations): http://www.onlinemathlearning.com/vector-subtraction.html




http://hyperphysics.phy-astr.gsu.edu/hbase/vect.html#vec9

Click here to see a SAMPLE VECTOR PROBLEM walked through, including all steps (e.g. breaking into components, finding the direction of the resultant vector,etc.)

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WEBSITES:




Very quick video illustrating how to use dimensional analysis to find the correct equation: https://www.youtube.com/watch?v=rLAEc0xcI2k



Vector Basics - Drawing Vectors/Vector Addition




Adding & Subtracting Vectors: https://www.youtube.com/watch?v=2dHk_yJ9ntQ&feature=player_embedded



Adding & Subtracting Vectors (part two): https://www.youtube.com/watch?v=mFu5IPOG5Cw



Introduction to Vectors:
https://www.youtube.com/watch?feature=player_embedded&v=-3E9SdgW8KU

*Note: this video describes the concept of equal vectors from 2:10-3:15 which we did not discuss in class, but the rest of the video is relevant





How to find the resultant of three or more vectors: https://www.youtube.com/watch?v=g_TnqKX5ybY

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SOURCES:
http://www.physicsclassroom.com/class/vectors/u3l1a.cfm

http://www.compadre.org/introphys/items/detail.cfm?ID=7782

http://sdsu-physics.org/physics180/physics195/Topics/chapter3.html

http://zonalandeducation.com/mstm/physics/mechanics/vectors/findingComponents/findingComponents.htm

http://www.numeracy-bank.net/images/trigonometric_functions/sctohanoa.jpg

http://www.physicsclassroom.com/Class/vectors/u3l1b3.gif

http://www.physchem.co.za/vectors/graphics/addition_f13d.gif

http://www.physchem.co.za/vectors/graphics/addition_f24.gif

http://www.physicsclassroom.com/class/vectors/u3l3b.cfm